Question

7) A random sample of 40 men were asked how if they enjoyed watching sports more than movies and 64% percent reported that they did. What is the 95% confidence interval of the true proportion of men who enjoy watching sports more than movies?

Answer 1

0.4913 < p < 0.7887

Explanation:

See the attached picture for detailed answer.

Answer 2

The 95% confidence interval of the true proportion of men who enjoy watching sports more than movies is (0.4912, 0.7888).

Explanation:

In a sample with a number n of people surveyed with a probability of a success of
`\pi`, and a confidence level of
`1-\alpha`, we have the following confidence interval of proportions.

`\pi \pm z\sqrt{(\pi(1-\pi))/(n)}`

In which

z is the zscore that has a pvalue of
`1 - (\alpha)/(2)`.

For this problem, we have that:

`n = 40, p =0.64`

95% confidence level

So
`\alpha = 0.05`, z is the value of Z that has a pvalue of
`1 - (0.05)/(2) = 0.975`, so
`Z = 1.96`.

The lower limit of this interval is:

`\pi - z\sqrt{(\pi(1-\pi))/(n)} = 0.64 - 1.96\sqrt{(0.64*0.36)/(40)} = 0.4912`

The upper limit of this interval is:

`\pi + z\sqrt{(\pi(1-\pi))/(n)} = 0.64 + 1.96\sqrt{(0.64*0.36)/(40)} = 0.7888`

The 95% confidence interval of the true proportion of men who enjoy watching sports more than movies is (0.4912, 0.7888).